Near the qualitative behavior depends upon whether or not
there is a ``resonance'' there. If there is, then
can begin with a complex component that
attenuates the propagation of EM energy in a (nearly static) applied
electric field. This (as we shall see) accurately describes conduction and resistance. If there isn't, then is
nearly all real and the material is a dielectric insulator.

Suppose there are both ``free'' electrons (counted by ) that are
``resonant'' at zero frequency, and ``bound'' electrons (counted by
). Then if we start out with:

(9.117)

where is now only the contribution from all the
``bound'' dipoles.

We can understand this from

(9.118)

(Maxwell/Ampere's Law). Let's first of all think of this in terms of a
plain old static current, sustained according to Ohm's Law:

(9.119)

If we assume a harmonic time dependence and a ``normal'' dielectric
constant , we get:

(9.120)

On the other hand, we can instead set the static current to zero
and consider all ``currents'' present to be the result of the
polarization response
to the field
. In this case:

(9.121)

Equating the two latter terms in the brackets and simplifying, we
obtain the following relation for the conductivity:

(9.122)

This is the Drude Model with the number of ``free''
electrons per unit volume. It is primarily useful for the insight that
it gives us concerning the ``conductivity'' being closely related to the
zero-frequency complex part of the permittivity. Note that at it is purely real, as it should be, recovering the usual Ohm's Law.

We conclude that the distinction between dielectrics and conductors is a
matter of perspective away from the purely static case. Away from the static
case, ``conductivity'' is simply a feature of resonant amplitudes. It is a
matter of taste whether a description is better made in terms of dielectric
constants and conductivity or complex dielectric.